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The elementary lecture consists of basics of TVM. It is easy to understand and interesting for novices.learn FINANCE, Make the sense of MONEY.
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JAHANGIR ALAM E-Mail: [email protected]
INTRODUCTIONTime Value of Money (TVM) is an important concept in financial management. It can be used
to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings,and annuities.
TVM is based on the concept that a dollar that you have today is worth more than the promise orexpectation that you will receive a dollar in the future. Money that you hold today is worth morebecause you can invest it and earn interest. After all, you should receive some compensation forforegoing spending. For instance, you can invest your dollar for one year at a 6% annual interest rateand accumulate $1.06 at the end of the year. You can say that the future value of the dollar is $1.06given a 6% interest rate and a one-year period. It follows that the present value of the $1.06 you expectto receive in one year is only $1.
A key concept of TVM is that a single sum of money or a series of equal, evenly-spaced payments orreceipts promised in the future can be converted to an equivalent value today. Conversely, you candetermine the value to which a single sum or a series of future payments will grow to at some futuredate.
You can calculate the fifth value if you are given any four of: Interest Rate, Number of Periods,Payments, Present Value, and Future Value. Each of these factors is very briefly defined in the right-hand column below. The left column has references to more detailed explanations, formulas, andexamples.
InterestInterest is the cost of borrowing money. An interest rate is the cost stated as a percent of the
amount borrowed per period of time, usually one year. The prevailing market rate is composed of:
1. The Real Rate of Interest that compensates lenders for postponing their own spending duringthe term of the loan.
2. An Inflation Premium to offset the possibility that inflation may erode the value of the moneyduring the term of the loan. A unit of money (dollar, peso, etc) will purchase progressivelyfewer goods and services during a period of inflation, so the lender must increase the interestrate to compensate for that loss..
3. Various Risk Premiums to compensate the lender for risky loans such as those that areunsecured, made to borrowers with questionable credit ratings, or illiquid loans that the lendermay not be able to readily resell.
The first two components of the interest rate listed above, the real rate of interest and an inflationpremium, collectively are referred to as the nominal risk-free rate. In the USA, the nominal risk-freerate can be approximated by the rate of US Treasury bills since they are generally considered to have avery small risk.
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Simple InterestSimple interest is calculated on the original principal only. Accumulated interest from prior
periods is not used in calculations for the following periods. Simple interest is normally used for asingle period of less than a year, such as 30 or 60 days.
Simple Interest = p * i * n
where: p = principal (original amount borrowed or loaned) i = interest rate for one period n = number of periods
Example 1: You borrow $10,000 for 3 years at 5% simple annual interest.interest = p * i * n = 10,000 * .05 * 3 = 1,500
Example 2: You borrow $10,000 for 60 days at 5% simple interest per year (assume a 365 day year).interest = p * i * n = 10,000 * .05 * (60/365) = 82.1917
Compound InterestCompound interest is calculated each period on the original principal and all interest
accumulated during past periods. Although the interest may be stated as a yearly rate, thecompounding periods can be yearly, semiannually, quarterly, or even continuously.
You can think of compound interest as a series of back-to-back simple interest contracts. The interestearned in each period is added to the principal of the previous period to become the principal for thenext period. For example, you borrow $10,000 for three years at 5% annual interest compoundedannually:interest year 1 = p * i * n = 10,000 * .05 * 1 = 500interest year 2 = (p2 = p1 + i1) * i * n = (10,000 + 500) * .05 * 1 = 525interest year 3 = (p3 = p2 + i2) * i * n = (10,500 + 525) *.05 * 1 = 551.25
Total interest earned over the three years = 500 + 525 + 551.25 = 1,576.25.Compare this to 1,500 earned over the same number of years using simple interest.
The power of compounding can have an astonishing effect on the accumulation of wealth. This tableshows the results of making a one-time investment of $10,000 for 30 years using 12% simpleinterest, and 12% interest compounded yearly and quarterly.
Type of Interest Principal Plus Interest Earned
Simple 46,000.00
Compounded Yearly 299,599.22
Compounded Quarterly 347,109.87
You can solve a variety of compounding problems including leases, loans, mortgages, and annuities byusing the present value, future value, present value of an annuity, and future value of an annuityformulas. See the index page to determine which of these is appropriate for your situation.
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Rate of ReturnWhen we know the Present Value (amount today), Future Value (amount to which the
investment will grow), and Number of Periods, we can calculate the rate of return with this formula:
i = ( FV / PV) (1/n) -1
In the table above we said that the Present Value is $10,000, the Future Value is $299,599.22, andthere are 30 periods. Confirm that the annual compound interest rate is 12%.
FV = 299,599.22PV = 10,000n = 30i = (299,599.22 / 10,000) 1/30 - 1 = 29.959922 .0333 - 1 = .12
Effective Rate (Effective Yield)The effective rate is the actual rate that you earn on an investment or pay on a loan after the
effects of compounding frequency are considered. To make a fair comparison between two interestrates when different compounding periods are used, you should first convert both nominal (or stated)rates to their equivalent effective rates so the effects of compounding can be clearly seen.The effective rate of an investment will always be higher than the nominal or stated interest rate wheninterest is compounded more than once per year. As the number of compounding periods increases, thedifference between the nominal and effective rates will also increase.To convert a nominal rate to an equivalent effective rate:
Effective Rate = (1 + (i / n))n - 1
Where: i = Nominal or stated interest rate n = Number of compounding periods per yearExample: What effective rate will a stated annual rate of 6% yield when compounded semiannually?Effective Rate = ( 1 + .06 / 2 )2 - 1 = .0609
Interest Rate in Calculations· Time Value of Money calculations always use compound interest.· You must adjust the interest rate and the number of periods to be consistent with compounding
periods. For example a 6% interest rate compounded semiannually for five years should beentered 3% (6 / 2) for 10 (5 * 2) periods.
· A calculator expects a 6% interest rate to be entered as the whole number 6 whereas formulastypically use a decimal value of .06.
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Number of PeriodsThe variable n in Time Value of Money formulas represents the number of periods. It is
intentionally not stated in years since each interval must correspond to a compounding period for asingle amount or a payment period for an annuity.
The interest rate and number of periods must both be adjusted to reflect the number of compoundingperiods per year before using them in TVM formulas. For example, if you borrow $1,000 for 2 yearsat 12% interest compounded quarterly, you must divide the interest rate by 4 to obtain rate of interestper period (i = 3%). You must multiply the number of years by 4 to obtain the total number of periods( n = 8).You can determine the number of periods required for an initial investment to grow to a specifiedamount with this formula:
number of periods = natural log [(FV * i) / (PV * i)] / natural log (1 + i)
where: PV = present value, the amount you invested FV = future value, the amount your investment will grow to i = interest per period
Example: You put $10,000 into a savings account at a 9.05% annual interest rate compoundedannually. How long will it take to double your investment?LN [(20,000 * .0905) / (10,000 * .0905)] / LN (1 .0905) =LN (2) / LN (1.0905) =.69314 / . 08663 = 8 years
Annuity PaymentsPayments in Time Value of Money formulas are a series of equal, evenly-spaced cash flows of
an annuity such as payments for a mortgage or monthly receipts from a retirement account.Payments must:
· be the same amount each period· occur at evenly spaced intervals· occur exactly at the beginning or end of each period· be all inflows or all outflows (payments or receipts)· represent the payment during one compounding (or discount) period
Calculate Payments When Present Value Is KnownThe Present Value is an amount that you have now, such as the price of property that you have justpurchased or the value of equipment that you have leased. When you know the present value, interestrate, and number of periods of an ordinary annuity, you can solve for the payment with this formula:
payment = PVoa / [(1- (1 / (1 + i)n )) / i]
Where: PVoa = Present Value of an ordinary annuity (payments are made at the end of each period) i = interest per period n = number of periods
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Example: You can get a $150,000 home mortgage at 7% annual interest rate for 30 years. Paymentsare due at the end of each month and interest is compounded monthly. How much will your paymentsbe?
PVoa = 150,000, the loan amounti = .005833 interest per month (.07 / 12)n = 360 periods (12 payments per year for 30 years)payment = 150,000 / [(1 - ( 1 / (1.005833)360)) / .005833] = 997.95
Calculate Payments When Future Value Is KnownThe Future Value is an amount that you wish to have after a number of periods have passed. Forexample, you may need to accumulate $20,000 in ten years to pay for college tuition. When you knowthe future value, interest rate, and number of periods of an ordinary annuity, you can solve for thepayment with this formula:
payment = FVoa / [((1 + i)n - 1 ) / i]
Where: FVoa = Future Value of an ordinary annuity (payments are made at the end of each period) i = interest per period n = number of periods
Example: In 10 years, you will need $50,000 to pay for college tuition. Your savings account pays5% interest compounded monthly. How much should you save each month to reach your goal?FVoa = 50,000, the future savings goali = .004167 interest per month (.05 / 12)n = 120 periods (12 payments per year for 10 years)payment = 50,000 / [(1.004167120 - 1) / .004167] = 321.99
Present ValuePresent Value Of A Single Amount
Present Value is an amount today that is equivalent to a future payment, or series of payments,that has been discounted by an appropriate interest rate. Since money has time value, the present valueof a promised future amount is worth less the longer you have to wait to receive it. The differencebetween the two depends on the number of compounding periods involved and the interest(discount) rate.The relationship between the present value and future value can be expressed as:
PV = FV [ 1 / (1 + i)n ]
Where:PV = Present ValueFV = Future Valuei = Interest Rate Per Periodn = Number of Compounding Periods
Example: You want to buy a house 5 years from now for $150,000. Assuming a 6% interest ratecompounded annually, how much should you invest today to yield $150,000 in 5 years?
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FV = 150,000i =.06n = 5PV = 150,000 [ 1 / (1 + .06)5 ] = 150,000 (1 / 1.3382255776) = 112,088.73
End of Year 1 2 3 4 5
Principal 112,088.73 118,814.05 125,942.89 133,499.46 141,509.43
Interest 6,725.32 7,128.84 7556.57 8,009.97 8,490.57
Total 118,814.05 125,942.89 133,499.46 141,509.43 150,000.00
Example 2: You find another financial institution that offers an interest rate of 6% compoundedsemiannually. How much less can you deposit today to yield $150,000 in five years?Interest is compounded twice per year so you must divide the annual interest rate by two to obtain arate per period of 3%. Since there are two compounding periods per year, you must multiply thenumber of years by two to obtain the total number of periods.
FV = 150,000i = .06 / 2 = .03n = 5 * 2 = 10PV = 150,000 [ 1 / (1 + .03)10] = 150,000 (1 / 1.343916379) = 111,614.09
Present Value of AnnuitiesAn annuity is a series of equal payments or receipts that occur at evenly spaced intervals.
Leases and rental payments are examples. The payments or receipts occur at the end of each periodfor an ordinary annuity while they occur at the beginning of each period.for an annuity due.
Present Value of an Ordinary AnnuityThe Present Value of an Ordinary Annuity (PVoa) is the value of a stream of expected or
promised future payments that have been discounted to a single equivalent value today. It is extremelyuseful for comparing two separate cash flows that differ in some way.PV-oa can also be thought of as the amount you must invest today at a specific interest rate so thatwhen you withdraw an equal amount each period, the original principal and all accumulated interestwill be completely exhausted at the end of the annuity.The Present Value of an Ordinary Annuity could be solved by calculating the present value of eachpayment in the series using the present value formula and then summing the results. A more directformula is:
PVoa = PMT [(1 - (1 / (1 + i)n)) / i]
Where:PVoa = Present Value of an Ordinary AnnuityPMT = Amount of each paymenti = Discount Rate Per Periodn = Number of Periods
Example 1: What amount must you invest today at 6% compounded annually so that you canwithdraw $5,000 at the end of each year for the next 5 years?
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PMT = 5,000i = .06n = 5PVoa = 5,000 [(1 - (1/(1 + .06)5)) / .06] = 5,000 (4.212364) = 21,061.82
Year 1 2 3 4 5
Begin 21,061.82 17,325.53 13,365.06 9,166.96 4,716.98
Interest 1,263.71 1,039.53 801.90 550.02 283.02
Withdraw -5,000 -5,000 -5,000 -5,000 -5,000
End 17,325.53 13,365.06 9,166.96 4,716.98 .00
Example 2: In practical problems, you may need to calculate both the present value of an annuity (astream of future periodic payments) and the present value of a single future amount:For example, a computer dealer offers to lease a system to you for $50 per month for two years. At theend of two years, you have the option to buy the system for $500. You will pay at the end of eachmonth. He will sell the same system to you for $1,200 cash. If the going interest rate is 12%, which isthe better offer?You can treat this as the sum of two separate calculations:
1. the present value of an ordinary annuity of 24 payments at $50 per monthly period Plus2. the present value of $500 paid as a single amount in two years.
PMT = 50 per period
i = .12 /12 = .01 Interest per period (12% annual rate / 12 payments per year)n = 24 number of periodsPVoa = 50 [ (1 - ( 1/(1.01)24)) / .01] = 50 [(1- ( 1 / 1.26973)) /.01] = 1,062.17
+FV = 500 Future value (the lease buy out)i = .01 Interest per periodn = 24 Number of periodsPV = FV [ 1 / (1 + i)n ] = 500 ( 1 / 1.26973 ) = 393.78
The present value (cost) of the lease is $1,455.95 (1,062.17 + 393.78). So if taxes are not considered,you would be $255.95 better off paying cash right now if you have it.
Present Value of an Annuity Due (PVad)The Present Value of an Annuity Due is identical to an ordinary annuity except that each
payment occurs at the beginning of a period rather than at the end. Since each payment occurs oneperiod earlier, we can calculate the present value of an ordinary annuity and then multiply the resultby (1 + i).
PVad = PVoa (1+i)
Where:PV-ad = Present Value of an Annuity DuePV-oa = Present Value of an Ordinary Annuityi = Discount Rate Per Period
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Example: What amount must you invest today a 6% interest rate compounded annually so that youcan withdraw $5,000 at the beginning of each year for the next 5 years?PMT = 5,000i = .06n = 5PVoa = 21,061.82 (1.06) = 22,325.53
Year 1 2 3 4 5
Begin 22,325.53 18,365.06 14,166.96 9,716.98 5,000.00
Interest 1,039.53 801.90 550.02 283.02
Withdraw -5,000.00 -5,000.00 -5,000.00 -5,000.00 -5,000.00
End 18,365.06 14,166.96 9,716.98 5,000.00 .00
Combined FormulaYou can also combine these formulas and the present value of a single amount formula into one.
Future ValueFuture Value Of A Single AmountFuture Value is the amount of money that an investment made today (the present value) will grow toby some future date. Since money has time value, we naturally expect the future value to be greaterthan the present value. The difference between the two depends on the number of compoundingperiods involved and the going interest rate.The relationship between the future value and present value can be expressed as:
FV = PV (1 + i)n
Where:FV = Future ValuePV = Present Valuei = Interest Rate Per Periodn = Number of Compounding Periods
Example: You can afford to put $10,000 in a savings account today that pays 6% interestcompounded annually. How much will you have 5 years from now if you make no withdrawals?PV = 10,000i = .06n = 5FV = 10,000 (1 + .06)5 = 10,000 (1.3382255776) = 13,382.26
End of Year 1 2 3 4 5
Principal 10,000.00 10,600.00 11,236.00 11,910.16 12,624.77
Interest 600.00 636.00 674.16 714.61 757.49
Total 10,600.00 11,236.00 11,910.16 12,624.77 13,382.26Example 2: Another financial institution offers to pay 6% compounded semiannually. How much
will your $10,000 grow to in five years at this rate?
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Interest is compounded twice per year so you must divide the annual interest rate by two to obtain arate per period of 3%. Since there are two compounding periods per year, you must multiply thenumber of years by two to obtain the total number of periods.PV = 10,000i = .06 / 2 = .03n = 5 * 2 = 10FV = 10,000 (1 + .03)10 = 10,000 (1.343916379) = 13,439.16
Future Value of AnnuitiesAn annuity is a series of equal payments or receipts that occur at evenly spaced intervals.
Leases and rental payments are examples. The payments or receipts occur at the end of each periodfor an ordinary annuity while they occur at the beginning of each period.for an annuity due.Future Value of an Ordinary AnnuityThe Future Value of an Ordinary Annuity (FVoa) is the value that a stream of expected or promisedfuture payments will grow to after a given number of periods at a specific compounded interest.The Future Value of an Ordinary Annuity could be solved by calculating the future value of eachindividual payment in the series using the future value formula and then summing the results. A moredirect formula is:
FVoa = PMT [((1 + i)n - 1) / i]
Where:FVoa = Future Value of an Ordinary AnnuityPMT = Amount of each paymenti = Interest Rate Per Periodn = Number of Periods
Example: What amount will accumulate if we deposit $5,000 at the end of each year for the next 5years? Assume an interest of 6% compounded annually.PV = 5,000i = .06n = 5FVoa = 5,000 [ (1.3382255776 - 1) /.06 ] = 5,000 (5.637092) = 28,185.46
Year 1 2 3 4 5
Begin 0 5,000.00 10,300.00 15,918.00 21,873.08
Interest 0 300.00 618.00 955.08 1,312.38
Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00
End 5,000.00 10,300.00 15,918.00 21,873.08 28,185.46
Example 2: In practical problems, you may need to calculate both the future value of an annuity (astream of future periodic payments) and the future value of a single amount that you have today:For example, you are 40 years old and have accumulated $50,000 in your savings account. You canadd $100 at the end of each month to your account which pays an interest rate of 6% per year. Willyou have enough money to retire in 20 years?
You can treat this as the sum of two separate calculations:1. the future value of 240 monthly payments of $100 Plus2. the future value of the $50,000 now in your account.
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PMT = $100 per periodi = .06 /12 = .005 Interest per period (6% annual rate / 12 payments per year)n = 240 periodsFVoa = 100 [ (3.3102 - 1) /.005 ] = 46,204 +PV = 50,000 Present value (the amount you have today)i = .005 Interest per periodn = 240 Number of periodsFV = PV (1+i)240 = 50,000 (1.005)240 = 165,510.22After 20 years you will have accumulated $211,714.22 (46,204.00 + 165,510.22).
Future Value of an Annuity Due (FVad)The Future Value of an Annuity Due is identical to an ordinary annuity except that each
payment occurs at the beginning of a period rather than at the end. Since each payment occurs oneperiod earlier, we can calculate the present value of an ordinary annuity and then multiply the resultby (1 + i).
FVad = FVoa (1+i)
Where:FVad = Future Value of an Annuity DueFVoa = Future Value of an Ordinary Annuityi = Interest Rate Per Period
Example: What amount will accumulate if we deposit $5,000 at the beginning of each year for thenext 5 years? Assume an interest of 6% compounded annually.PV = 5,000i = .06n = 5FVoa = 28,185.46 (1.06) = 29,876.59
Year 1 2 3 4 5
Begin 0 5,300.00 10,918.00 16,873.08 23,185.46
Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00
Interest 300.00 618.00 955.08 1,312.38 1,691.13
End 5,300.00 10,918.00 16,873.08 23,185.46 29,876.59
Combined FormulaYou can also combine these formulas and the future value of a single amount formula into one.
Loan AmortizationAmortization
Amortization is a method for repaying a loan in equal installments. Part of each payment goestoward interest due for the period and the remainder is used to reduce the principal (the loan balance).As the balance of the loan is gradually reduced, a progressively larger portion of each payment goestoward reducing principal.For Example, the 15 and 30 year fixed-rate mortgages common in the US are fully amortized loans. Topay off a $100,000, 15 year, 7%, fixed-rate mortgage, a person must pay $898.83 each month for 180months (with a small adjustment at the end to account for rounding). $583.33 of the first payment goestoward interest and $315.50 is used to reduce principal. But by payment 179, only $10.40 is needed for
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interest and $888.43 is used to reduce principal.
Amortization ScheduleAn amortization schedule is a table with a row for each payment period of an amortized loan.
Each row shows the amount of the payment that is needed to pay interest, the amount that is used toreduce principal, and the balance of the loan remaining at the end of the period.The first and last 5 months of an amortization schedule for a $100,000, 15 year, 7%, fixed-ratemortgage will look like this:
Amortization Schedule
Month Principal Interest Balance
1 -315.50 -583.33 99,684.51
2 -317.34 -581.49 99,367.17
3 -319.19 -579.64 99,047.98
4 -321.05 -577.78 98,726.93
5 -322.92 -575.91 98,404.01
Rows 6-175 omitted
176 -873.07 -25.76 3,543.48
177 -878.16 -20.67 2,665.32
178 -883.28 -15.55 1,782.04
179 -888.43 -10.40 893.61
180 -893.62 -5.21 -0.01Negative Amortization
Negative amortization occurs when the payment is not large enough to cover the interest duefor a period. This will cause the loan balance to increase after each payment - a situation that shouldcertainly be avoided. This might occur, for instance, if the rate of an adjustable-rate loan increases, butthe payment does not.
Cash Flow DiagramsA cash flow diagram is a picture of a financial problem that shows all cash inflows and
outflows plotted along a horizontal time line. It can help you to visualize a financial problem and todetermine if it can be solved using TVM methods.
Constructing a CashFlow DiagramThe time line is a horizontal line divided into equal periods such as days, months, or
years. Each cash flow, such as a payment or receipt, is plotted along this line at the beginning or endof the period in which it occurs. Funds that you pay out such as savings deposits or lease paymentsare negative cash flows that are represented by arrows which extend downward from the time linewith their bases at the appropriate positions along the line. Funds that you receive such as proceedsfrom a mortgage or withdrawals from a saving account are positive cash flows represented by arrowsextending upward from the line.
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Example: You are 40 years old and have accumulated $50,000 in your savings account. You can add$100 at the end of each month to your account which pays an annual interest rate of 6% compoundedmonthly. Will you be able to retire in 20 years?
The time line is divided into 240 monthly periods (20 years times 12 payments per year) since thepayments are made monthly and the interest is also compounded monthly. The $50,000 that you havenow (present value) is a negative cash outflow since you will treat it as though you were just nowdepositing it into the account. It is represented with a downward pointing arrow with its base at thebeginning of the first period. The 240 monthly $100 deposits are also negative outflows representedwith downward pointing arrows placed at the end of each period. Finally you will withdraw someunknown amount (the future value) after 20 years. Represent this positive inflow with an upwardpointing arrow with its base at the very end of the last period.This diagram was drawn from your point of view. From the bank's point of view, the present valueand the series of deposits are positive cash inflows, and the final withdrawal of the future value will bea negative outflow.See the Future Value of an Ordinary Annuity page for the solution to this problem.
Identifying TVM ProblemsFor a financial problem to be solved with time value of money formulas:
· the periods must be of equal length· payments, if present, must all be equal and be all inflows or all outflows· payments must all occur either at the beginning or end of a period· the interest rate cannot vary along the time line
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Acknowledgement:
The handout is prepared according to lecture sheet of various university & websearch. For any copyright disclaimer author will not be liable incase of further copying.It is prepared only study purpose not for business.
Jahangir AlamBBA (pursuing)
Department of ManagementUniversity of Rajshahi.
Bangladesh
